

with \(\big (f_k(t),\alpha ^{-1} g_k(t)\big ) \in \mathcal {L}^2\) .
Hence, by regularity theory for elliptic problems with bulk-surface coupling (see [1]}), we deduce that \(\big (\phi _k(t),\psi _k(t)\big ) \in \mathcal {H}^2\)  with
\(&\big \Vert  \big (\phi _k(t),\psi _k(t)\big ) \big \Vert _{\mathcal {H}^2}^2\le C \Vert  \big (f_k(t),\alpha ^{-1} g_k(t)\big ) \Vert _{\mathcal {L}^2}^2\\&\quad \le C \Big ( \Vert  \mu _k(t) \Vert _{L^2(\Omega )}^2+ \big \Vert  F^{\prime }\big (\phi _k(t)\big ) \big \Vert _{L^2(\Omega )}^2+ \Vert  \theta _k(t) \Vert _{L^2(\Gamma )}^2+ \big \Vert  G^{\prime }\big (\psi _k(t)\big ) \big \Vert _{L^2(\Gamma )}^2 \Big )\\&\quad \le C + C \Vert  \mu _k(t) \Vert _{L^2(\Omega )}^2 + C\Vert  \theta _k(t) \Vert _{L^2(\Gamma )}^2 + C\epsilon ^2 \big \Vert  \big (\phi _k(t),\psi _k(t)\big ) \big \Vert _{\mathcal {H}^2}^2\) 
